The concept of asymptotic correctability of Bell-diagonal quantum states is generalized to elementary quantum systems of higher dimensions. Based on these results basic properties of quantum state purification protocols are investigated which are capable of purifying tensor products of Bell-diagonal states and which are based on B -steps of the Gottesman-Lo-type with the subsequent application of a Calderbank–Shor–Steane quantum code. Consequences for maximum tolerable error rates of quantum cryptographic protocols are discussed.
The general conditions are discussed which quantum state purification protocols have to fulfil in order to be capable of purifying Bell-diagonal qubit-pair states, provided they consist of steps that map Bell-diagonal states to Bell-diagonal states and they finally apply a suitably chosen Calderbank–Shor–Steane code to the outcome of such steps. As a main result a condition on asymptotic correctability is presented, which relates this problem to the magnitude of a characteristic exponent governing the relation between bit and phase errors under the purification steps. This condition allows a straightforward determination of maximum tolerable bit-error rates of quantum key distribution protocols whose security analysis can be reduced to the purification of Bell-diagonal states.
Symmetric extendibility of quantum states has recently drawn attention in the context of quantum cryptography to judge whether quantum states shared between two distant parties can be purified by means of one-way error correction protocols. In this paper we study the symmetric extendibility in a specific class of two-qudit states, i.e., states composed of two
d-level systems, in order to find upper bounds on tolerable error rates for a wide class of qudit-based quantum cryptographic protocols using two-way error correction. In important cases these bounds coincide with previously known lower bounds, thereby proving sharpness of these bounds in arbitrary finite-dimensional systems.
Abstract : We consider quantum error-correcting codes over alphabets of arbitrary size. We generalize quantum stabilizer codes and develop Calderbank-Shor-Steane construction over quantum systems of arbitrary dimensions using the group structure of alphabets. We also develop a methodology systematically conjoining error-correcting codes into a new class of error-correcting codes. With the help of these methods we give several ways to construct quantum maximum distance separable (MDS) codes and present many families of quantum MDS codes.
This paper shows that quantum computation can be made fault-tolerant against errors and inaccuracies when
η, the probability for an error in a qubit or a gate, is smaller than a constant threshold
ηc. This result improves on Shor's result [Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE, Los Alamitos, CA, 1996, pp. 56–65], which shows how to perform fault-tolerant quantum computation when the error rate
η decays polylogarithmically with the size of the computation, an assumption which is physically unreasonable. The cost of making the quantum circuit fault-tolerant in our construction is polylogarithmic in time and space. Our result holds for a very general local noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault-tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with
p>2 states, namely, p-qudits, and is also generalized to one-dimensional quantum computers with only nearest-neighbor interactions. No measurements, or classical operations, are required during the quantum computation. We estimate the threshold of our construction to be
ηc≃10−6, in the best case. By this we show that local noise is in principle not an obstacle for scalable quantum computation. The main ingredient of our proof is the computation on states encoded by a quantum error correcting code (QECC). To this end we introduce a special class of Calderbank–Shor–Steane (CSS) codes, called polynomial codes (the quantum analogue of Reed–Solomon codes). Their nice algebraic structure allows all of the encoded gates to be transversal. We also provide another version of the proof which uses more general CSS codes, but its encoded gates are slightly less elegant. To achieve fault tolerance, we encode the quantum circuit by another circuit by using one of these QECCs. This step is repeated polyloglog many times, each step slightly improving the effective error rate, to achieve the desired reliability. The resulting circuit exhibits a hierarchical structure, and for the analysis of its robustness we borrow terminology from Khalfin and Tsirelson [Found. Phys., 22 (1992), pp. 879–948] and Gács [Advances in Computing Research: A Research Annual: Randomness and Computation, JAI Press, Greenwich, CT, 1989]. The paper is to a large extent self-contained. In particular, we provide simpler proofs for many of the known results we use, such as the fact that it suffices to correct for bit-flips and phase-flips, the correctness of CSS codes, and the fact that two-qubit gates are universal, together with their extensions to higher-dimensional particles. We also provide full proofs of the universality of the sets of gates we use (the proof of universality was missing in Shor's paper). This paper thus provides a self-contained and complete proof of universal fault-tolerant quantum computation in the presence of local noise.
The tolerable erasure error rate for scalable quantum computation is shown to be at least 0.292, given standard scalability assumptions. This bound is obtained by implementing computations with generic stabilizer code teleportation steps that combine the necessary operations with error correction. An interesting consequence of the technique is that the only errors that affect the maximum tolerable error rate are storage and Bell measurement errors. If storage errors are negligible, then any detected Bell measurement error below 1/2 is permissible. Another consequence of the technique is that the maximum tolerable depolarizing error rate is dominated by how well one can prepare the required encoded states. For example, if storage and Bell measurement errors are relatively small, then independent depolarizing errors with error rate close to 0.1 per qubit are tolerable in the prepared states. The implementation overhead is dominated by the efficiency with which the required encoded states can be prepared. At present, this efficiency is very low, particularly for error rates close to the maximum tolerable ones.